We characterize the group property of being with infinite conjugacy classes (or icc, i.e. infinite and of which all conjugacy classes except $\left\{1\right\}$ are infinite) for groups which are extensions of groups. We prove a general result for extensions of groups, then deduce characterizations in semi-direct products, wreath products, finite extensions, among others examples we also deduce a characterization for amalgamated products and HNN extensions. The icc property is correlated to the Theory of von Neumann...

This paper deals with a rationality condition for groups. Let n be a fixed positive integer. Suppose every element g of the finite solvable group is conjugate to its nth power g n. Let p be a prime divisor of the order of the group. We conclude that the multiplicative order of n modulo p is small, or p is small.

It is proved that if a locally soluble group of infinite rank has only finitely many non-trivial conjugacy classes of subgroups of infinite rank, then all its subgroups are normal.

Let F C 0 be the class of all finite groups, and for each nonnegative integer n define by induction the group class FC^(n+1) consisting of all groups G such that for every element x the factor group G/CG ( <x>^G ) has the property FC^n . Thus FC^1 -groups are precisely groups with finite conjugacy classes, and the class FC^n obviously contains all finite groups and all nilpotent groups with class at most n. In this paper the known theory of FC-groups is taken as a model, and it is shown that...

On présente des conditions suffisantes pour qu’une extension HNN soit intérieurement moyennable, respectivement CCI, qui donnent des critères nécessaires et suffisants parmi les groupes de Baumslag-Solitar. On en déduit qu’un tel groupe, vu comme groupe d’automorphismes de son arbre de Bass-Serre, possède des éléments non triviaux qui fixent des sous-arbres non bornés.

Let $G$ be a finite group. A normal subgroup $N$ of $G$ is a union of several $G$-conjugacy classes, and it is called $n$-decomposable in $G$ if it is a union of $n$ distinct $G$-conjugacy classes. In this paper, we first classify finite non-perfect groups satisfying the condition that the numbers of conjugacy classes contained in its non-trivial normal subgroups are two consecutive positive integers, and we later prove that there is no non-perfect group such that the numbers of conjugacy classes contained in its...

A group $G$ is said to be a PC-group, if $G/C_G\left(x^G\right)$ is a polycyclic-by-finite group for all $x\in G$. A minimal non-PC-group is a group which is not a PC-group but all of whose proper subgroups are PC-groups. Our main result is that a minimal non-PC-group having a non-trivial finite factor group is a finite cyclic extension of a divisible abelian group of finite rank.

In this article, we study the elements with disconnected centralizer in the Brauer complex associated to a simple algebraic group $\mathbf{G}$ defined over a finite field with corresponding Frobenius map $F$ and derive the number of $F$-stable semisimple classes of $\mathbf{G}$ with disconnected centralizer when the order of the fundamental group has prime order. We also discuss extendibility of semisimple characters of the fixed point subgroup ${\mathbf{G}}^F$ to their inertia group in the full automorphism group. As a consequence, we...

Let $N$ be a normal subgroup of a group $G$. The structure of $N$ is given when the $G$-conjugacy class sizes of $N$ is a set of a special kind. In fact, we give the structure of a normal subgroup $N$ under the assumption that the set of $G$-conjugacy class sizes of $N$ is $(p_{1n_1}^{a_{1n_1}},\cdots ,p_{11}^{a_{11}},1)\times \cdots \times (p_{r{n}_r}^{a_{r{n}_r}},\cdots ,p_{r1}^{a_{r1}},1)$, where $r>1$, $n_i>1$ and $p_{ij}$ are distinct primes for $i\in \{1,2,\cdots ,r\}$, $j\in \{1,2,\cdots ,n_i\}$.

Gli autori studiano il sottogruppo $\mathrm{\Sigma }\left(G\right)$ intersezione dei sottogruppi massimali e non supersolubili di un gruppo finito $G$ e le relazioni tra la struttura di $G$ e quella di $\mathrm{\Sigma }\left(G\right)$.

For a finite group $G$ and a non-linear irreducible complex character $\chi$ of $G$ write $\upsilon \left(\chi \right)=\{g\in G\mid \chi (g)=0\}$. In this paper, we study the finite non-solvable groups $G$ such that $\upsilon \left(\chi \right)$ consists of at most two conjugacy classes for all but one of the non-linear irreducible characters $\chi$ of $G$. In particular, we characterize a class of finite solvable groups which are closely related to the above-mentioned question and are called solvable $\varphi$-groups. As a corollary, we answer Research Problem $2$ in [Y. Berkovich and L. Kazarin: Finite...

For the groups $GU\left(3\right)$, $SU\left(3\right)$, $GL\left(3\right)$, $SU\left(3\right)$ over a finite field we solve the class product problem, i.e., we give a complete list of $m$-tuples of conjugacy classes whose product does not contain the identity matrix.